Society Collapse through erroneous Annual Tax rates: Piketty Recipe
A simple $N$-agents dynamic model is introduced with two ingredients:
I) During the year, agents are randomly tossed to increase their wealth. Tossed agent $i$ doubles its wealth $W_i$ towards $2W_i$. The tossing is repeated $N$ times, one agent could be tossed more than once. Then, the wealth distribution is kept normalized $\sum_i W_i=1$.
II) At the year end, all agents pay their annual taxes with rates $pW_i$ (net tax $pW_i^2$), where $p$ is a control parameter. Then, the wealth distribution is again kept normalized, $\sum_i W_i=1$.
(As a matter of technicality, the normalization is performed because we are not interested in the economy growing as a whole, only in the distribution of wealths and its inequalities. For those interested in it, we can inform that the factor 2 warrants an forever-growing economy.)
For $p=0$, all agents pay the same tax rate. After many years, the whole society collapses, i.e. a single agent owns the whole wealth. This situation is an absorbing state. For $p>0$, rich agents pay larger taxes than poor agents. For $p$ larger than a critical value $p_c \approx 0.3$ the collapse is avoided, the distribution of wealths survives forever and the absorbing state is never reached. So, there is a transition at $p_c$. An order parameter for this transition is constructed by ordering the wealth distribution in decreasing order (Zipf plot), and calculating its first moment. The result is obviously null for $p<p_c$ (collapse), but positive in the surviving phase $p>p_c$. We are now calculating the precise value of $p_c$, as well as the critical indices of this transition.
We tested also the Tea Party ideology, rich agents paying smaller tax rates than poor agents. This corresponds to $p<0$. The result is simply a predictable collapse acceleration, according to Piketty.